#Cod Octave:
pkg load symbolic;
syms m;
v1=[1,3,2]; v2=[1,-2,3]; v3=[0,5,m];
# valoarea lui m
m = solve(det([v1.',v2.',v3.'])==0, m)
#relatia de dependenta (fractii)
format rat;
rref([v1.',v2.',v3.',[0;0;0]])
# test
# reinitializare vectori cu m calculat
v1=[1,3,2]; v2=[1,-2,3]; v3=[0,5,m];
# verificare ecuatii
v1 - v2 - v3 # ar trebui sa returneze (0,0,0)
#Cod Octave:
pkg load symbolic;
syms m;
v1=[1,4,5]; v2=[-1,3,m]; v3=[2,-1,1];
# valoarea lui m
m = solve(det([v1.',v2.',v3.'])==0, m)
#relatia de dependenta (fractii)
format rat;
rref([v1.',v2.',v3.',[0;0;0]])
# test
# reinitializare vectori cu m calculat
v1=[1,4,5]; v2=[-1,3,m]; v3=[2,-1,1];
# verificare ecuatii
-5*v1 + 9*v2 + 7*v3 # ar trebui sa returneze (0,0,0)
# Cod Octave
A = [2,2,3; 1,-1,0; -1,2,1];
det(A) # diferit de 0 => A este inversabila
format rat;
inv_A = inv(A)
sum(inv_A(3,:)) # ans = 9
# Cod Octave
A = [2,3,1; 3,6,2; 1,2,1];
det(A) # diferit de 0 => A este inversabila
format rat;
inv_A = inv(A)
sum(inv_A(3,:)) # ans = 2
# Cod Octave
A = [1,0,1; 0,1,1; 1,1,1];
det(A) # diferit de 0 => A este inversabila
format rat;
inv_A = inv(A)
sum(inv_A(1,:)) # ans = 0
# Cod Octave
A = [2,2,-1; 2,-1,2; -1,2,2];
det(A) # diferit de 0 => A este inversabila
format rat;
inv_A = inv(A)
det(inv_A) # ans = -1/27
# Cod Octave
A = [1,1,-1; 3,-2,2; 2,3,-2];
det(A) # diferit de 0 => A este inversabila
format rat;
inv_A = inv(A)
det(inv_A) # ans = -1/5
#Cod Octave:
M = [6,5;-3,-3];
det(M^2)
#Cod Octave:
M = [4,3;-6,-5];
det(M^2)
#Cod Octave:
M = [3,-2;-2,3];
det(M^2)
# Cod Octave:
pkg load symbolic;
A = sym([4,1; -2,1]);
syms lambda;
charpoly(A, lambda) # ans = lambda^2 -5*lambda + 6
# Cod Octave:
pkg load symbolic;
A = sym([-4,-3; -2,-5]);
syms lambda;
charpoly(A, lambda) # ans = lambda^2 + 9*lambda + 14
# Cod Octave:
pkg load symbolic;
A = sym([-2,1; -4,3]);
syms lambda;
charpoly(A, lambda) # ans = lambda^2 - lambda - 2
# Cod Octave:
A = [3,2,5,4; 2,1,3,3; 1,2,3,0];
rank(A) # ans = 2
# Cod Octave:
A = [2,-1,1,2; 1,1,2,1; 3,-2,1,3];
rank(A) # ans = 2
# Cod Octave:
A = [1,-2,0,1; 3,-1,-2,0; 2,1,-2,-1];
rank(A) # ans = 2
Cod Octave:
pkg load symbolic;
syms m;
M = [[1,2,-1];[4,-1,3];[5,1,m]];
solve(det(M)==0, m)
Cod Octave:
pkg load symbolic;
syms m;
M = [[2,-1,3];[1,2,-4];[-3,4,m]];
solve(det(M)==0, m)
Cod Octave:
pkg load symbolic;
syms m;
M = [[-1,1,0];[2,-2,1];[1,m,1]];
solve(det(M)==0, m)
# Cod Octave:
A = [4,-3; 1,-1];
det(A) # ans = -1
# Cod Octave:
A = [8,3; 4,2];
det(A) # ans = 4
# Cod Octave:
A = [3,1; 5,2];
det(A) # ans = 1
# Cod Octave
A = [-3,7; -2,5];
inv_A = inv(A);
sum(inv_A'(:)) # ans = 3
# Cod Octave
A = [5,8; 2,3];
inv_A = inv(A);
sum(inv_A'(:)) # ans = 2
# Cod Octave
A = [3,1; 5,2];
inv_A = inv(A);
sum(inv_A'(:)) # ans = -1
Cod Octave:
M = [[2,-3,1,0];[1,5,-3,0];[5,12,-8,0]]; rank(M)
Cod Octave:
M = [[1,1,-1,0];[1,-1,1,0];[5,-1,1,0]]; rank(M)
# Cod Octave:
v1=[2,1,-1]; v2=[1,-1,2]; v3=[4,-1,3];
rref([v1.',v2.',v3.',[0;0;0]])
# test
v1 + 2*v2 -v3 # ans = [0,0,0]
# Cod Octave:
v1=[2,-5,3]; v2=[3,-8,1]; v3=[1,-2,5];
rref([v1.',v2.',v3.',[0;0;0]])
# test
2*v1 - v2 - v3 # ans = [0,0,0]
# Cod Octave:
v1=[2,-5,3]; v2=[3,-8,1]; v3=[1,-2,5];
M = [v1.',v2.',v3.'];
rank(M) # ans = 2
# Cod Octave:
v1=[2,1,5]; v2=[-3,9,-6]; v3=[-5,1,-12];
M = [v1.',v2.',v3.'];
rank(M) # ans = 2
# Cod Octave:
pkg load symbolic;
syms lambda;
solve(det([[11-lambda,16];[-2,-1-lambda]])==0,lambda)
# Cod Octave:
pkg load symbolic;
syms lambda;
solve(det([[-4-lambda,-3];[-2,-5-lambda]])==0,lambda)
# Cod Octave:
pkg load symbolic;
syms lambda;
solve(det([[4-lambda,1];[-2,1-lambda]])==0,lambda)
# Cod Octave:
M = [[1,2,-2,-5];[-4,1,2,-1];[2,-1,3,9]];
sym(round(rref(M))(:,4))
# Cod Octave:
M = [[1,1,-1,0];[3,-2,2,5];[2,3,-2,2]];
sym(round(rref(M))(:,4))
# Cod Octave:
gcd(11432775, 15265170) # ans = 32205
#Cod Octave:
pkg load symbolic;
syms m;
v1=[1,3,2]; v2=[1,-2,3]; v3=[0,5,m];
# valoarea lui m
m = solve(det([v1.',v2.',v3.'])==0, m)
#relatia de dependenta (fractii)
format rat;
rref([v1.',v2.',v3.',[0;0;0]])
# test
# reinitializare vectori cu m calculat
v1=[1,3,2]; v2=[1,-2,3]; v3=[0,5,m];
# verificare ecuatii
v1 - v2 - v3 # ar trebui sa returneze (0,0,0)
#Cod Octave:
pkg load symbolic;
syms m;
v1=[1,4,5]; v2=[-1,3,m]; v3=[2,-1,1];
# valoarea lui m
m = solve(det([v1.',v2.',v3.'])==0, m)
#relatia de dependenta (fractii)
format rat;
rref([v1.',v2.',v3.',[0;0;0]])
# test
# reinitializare vectori cu m calculat
v1=[1,4,5]; v2=[-1,3,m]; v3=[2,-1,1];
# verificare ecuatii
-5*v1 + 9*v2 + 7*v3 # ar trebui sa returneze (0,0,0)
# Cod Octave
A = [2,2,3; 1,-1,0; -1,2,1];
det(A) # diferit de 0 => A este inversabila
format rat;
inv_A = inv(A)
sum(inv_A(3,:)) # ans = 9
# Cod Octave
A = [2,3,1; 3,6,2; 1,2,1];
det(A) # diferit de 0 => A este inversabila
format rat;
inv_A = inv(A)
sum(inv_A(3,:)) # ans = 2
# Cod Octave
A = [1,0,1; 0,1,1; 1,1,1];
det(A) # diferit de 0 => A este inversabila
format rat;
inv_A = inv(A)
sum(inv_A(1,:)) # ans = 0
# Cod Octave
A = [2,2,-1; 2,-1,2; -1,2,2];
det(A) # diferit de 0 => A este inversabila
format rat;
inv_A = inv(A)
det(inv_A) # ans = -1/27
# Cod Octave
A = [1,1,-1; 3,-2,2; 2,3,-2];
det(A) # diferit de 0 => A este inversabila
format rat;
inv_A = inv(A)
det(inv_A) # ans = -1/5
#Cod Octave:
M = [6,5;-3,-3];
det(M^2)
#Cod Octave:
M = [4,3;-6,-5];
det(M^2)
#Cod Octave:
M = [3,-2;-2,3];
det(M^2)
# Cod Octave:
pkg load symbolic;
A = sym([4,1; -2,1]);
syms lambda;
charpoly(A, lambda) # ans = lambda^2 -5*lambda + 6
# Cod Octave:
pkg load symbolic;
A = sym([-4,-3; -2,-5]);
syms lambda;
charpoly(A, lambda) # ans = lambda^2 + 9*lambda + 14
# Cod Octave:
pkg load symbolic;
A = sym([-2,1; -4,3]);
syms lambda;
charpoly(A, lambda) # ans = lambda^2 - lambda - 2
# Cod Octave:
A = [3,2,5,4; 2,1,3,3; 1,2,3,0];
rank(A) # ans = 2
# Cod Octave:
A = [2,-1,1,2; 1,1,2,1; 3,-2,1,3];
rank(A) # ans = 2
# Cod Octave:
A = [1,-2,0,1; 3,-1,-2,0; 2,1,-2,-1];
rank(A) # ans = 2
Cod Octave:
pkg load symbolic;
syms m;
M = [[1,2,-1];[4,-1,3];[5,1,m]];
solve(det(M)==0, m)
Cod Octave:
pkg load symbolic;
syms m;
M = [[2,-1,3];[1,2,-4];[-3,4,m]];
solve(det(M)==0, m)
Cod Octave:
pkg load symbolic;
syms m;
M = [[-1,1,0];[2,-2,1];[1,m,1]];
solve(det(M)==0, m)
# Cod Octave:
A = [4,-3; 1,-1];
det(A) # ans = -1
# Cod Octave:
A = [8,3; 4,2];
det(A) # ans = 4
# Cod Octave:
A = [3,1; 5,2];
det(A) # ans = 1
# Cod Octave
A = [-3,7; -2,5];
inv_A = inv(A);
sum(inv_A'(:)) # ans = 3
# Cod Octave
A = [5,8; 2,3];
inv_A = inv(A);
sum(inv_A'(:)) # ans = 2
# Cod Octave
A = [3,1; 5,2];
inv_A = inv(A);
sum(inv_A'(:)) # ans = -1
Cod Octave:
M = [[2,-3,1,0];[1,5,-3,0];[5,12,-8,0]]; rank(M)
Cod Octave:
M = [[1,1,-1,0];[1,-1,1,0];[5,-1,1,0]]; rank(M)
# Cod Octave:
v1=[2,1,-1]; v2=[1,-1,2]; v3=[4,-1,3];
rref([v1.',v2.',v3.',[0;0;0]])
# test
v1 + 2*v2 -v3 # ans = [0,0,0]
# Cod Octave:
v1=[2,-5,3]; v2=[3,-8,1]; v3=[1,-2,5];
rref([v1.',v2.',v3.',[0;0;0]])
# test
2*v1 - v2 - v3 # ans = [0,0,0]
# Cod Octave:
v1=[2,-5,3]; v2=[3,-8,1]; v3=[1,-2,5];
M = [v1.',v2.',v3.'];
rank(M) # ans = 2
# Cod Octave:
v1=[2,1,5]; v2=[-3,9,-6]; v3=[-5,1,-12];
M = [v1.',v2.',v3.'];
rank(M) # ans = 2
# Cod Octave:
pkg load symbolic;
syms lambda;
solve(det([[11-lambda,16];[-2,-1-lambda]])==0,lambda)
# Cod Octave:
pkg load symbolic;
syms lambda;
solve(det([[-4-lambda,-3];[-2,-5-lambda]])==0,lambda)
# Cod Octave:
pkg load symbolic;
syms lambda;
solve(det([[4-lambda,1];[-2,1-lambda]])==0,lambda)
# Cod Octave:
M = [[1,2,-2,-5];[-4,1,2,-1];[2,-1,3,9]];
sym(round(rref(M))(:,4))
# Cod Octave:
M = [[1,1,-1,0];[3,-2,2,5];[2,3,-2,2]];
sym(round(rref(M))(:,4))
# Cod Octave:
gcd(11432775, 15265170) # ans = 32205